Does Compressed Sensing bring anything new to data Compression?

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With regards to your question "...given that the data is already on the computer -- does compressed sensing offer anything? For example, would it offer better data compression? Would it result in better image search?..."

In general the answer to your question is no it would not offer better data compression at least initially! This is the case for images where nonlinear schemes like jpeg does better than compressed sensing by a constant of 4 to 5 and comes from the klog(N/K) constant found in diverse theoretical results in different papers.

I said initially because right now compressed sensing is mostly focused on the concept of sparsity but there is new work now coming up that tries to use additional information such as the fact that wavelets decomposition comes in clumps that could improve the compression. This work and others are likely to provide additional improvement with maybe the possibility of getting close to the nonlinear transform such as jpeg.

The other thing you have to keep in mind is that jpeg is the result of a focused effort of the whole industry and many years of research. So it really is difficult to do better than that but compressive sensing really provides some means of compression of other datasets without the need for the years of experience and manpower.

Finally, there is something immensely awe inspiring in the compression found in compressive sensing. It is universal, this means that right now you may "decode" image to a certain level of detail and then in ten years, using the same data you might actually "decode" a better image/dataset (this is with the caveat that the information was there in the first place) because your solvers will be better. You cannot do that with jpeg or jpeg2000 because the data that is compressed is intrinsically connected to the decoding scheme.

(disclosure: I write a small blog on compressed sensing)

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